M_p^3 = \frac{V_s^3}{2\pi G} PM_s^2Solve Binary Star Mass Homework Equation

[unscientific]

Homework Statement

How do I show that for a binary star system, if one star has mass $M_s$, speed $V_s$, period $P$, the mass of the other star is given by: $M_P^3 \approx \frac{V_s^3}{2\pi G} PM_s^2$?

Homework Equations

The Attempt at a Solution

\frac{GM_pM_s}{(a_p+a_s)^2} = \frac{M_s v_s^2}{a_s}
Substituting in $PV_s=2\pi a_s$:
M_p = \frac{2\pi(a_p+a_s)^2V_s}{PG}
Using kepler's second law: $P^2 = \frac{(a_p+a_s)^3(2\pi)^2}{G(M_p+M_s)}$:
M_p^3 = \frac{V_s^3}{2\pi G} P (M_p+M_s)^2

 

[Fightfish]

Seems correct to me. The mass of the planet is more often than not negligible compared to the mass of a star, so M_{p} << M_{S}

 

[unscientific]

Seems correct to me. The mass of the planet is more often than not negligible compared to the mass of a star, so M_{p} << M_{S}

Thanks a lot! I'm doing an introductory course to my first ever astrophysics module, so I'm not quite familiar with these things.